sum
with the same dimensions as the input matrix plus one extra row and column (for easier calculations).sum
matrix by iterating over the input matrix and updating sum[i+1][j+1]
to be the sum of matrix[i][j]
plus the values from the left and above, minus the value diagonally up-left to avoid double-counting.sumRegion
by applying the inclusion-exclusion principle to obtain the sum of the desired submatrix using the pre-computed sum
matrix.